Quantum Action-Dependent Channels

TL;DR

This work introduces the quantum action-dependent channel, where the transmitter's actions influence the channel environment via a quantum action channel, with side information implemented through entanglement due to the no-cloning constraint. It provides a one-shot achievable rate, derived via quantum information techniques such as pinching and sandwiched Rényi divergences, and shows that the rate scales as $R_{low} = I(VU;B)_{\rho} - I(V;S|U)_{\rho}$ after optimizing over the action and input encodings. The analysis blends a two-stage encoding approach (action encoding and message encoding) with a pinching-based decoding strategy, yielding rigorous non-asymptotic bounds. In the asymptotic limit, the paper proves that the quantum action-dependent channel capacity satisfies $C_{\text{QAD}} \ge \max_{p_{VU},\sigma_G^u,\mathcal{F}_{S_0\to A}^v}[I(VU;B)_{\rho} - I(V;S|U)_{\rho}]$, extending Weissman’s classical action-dependent framework to the quantum regime with entanglement-assisted side information. The results highlight the interplay between action-induced environmental control and quantum information measures in achieving reliable quantum communication with environment-dependent channels.

Abstract

We study the quantum action-dependent channel. The model can be viewed as a quantum analog of the classical action-dependent channel model. In this setting, the communication channel has two inputs: Alice's transmission and the input environment. The action-dependent mechanism enables the transmitter to influence the channel's environment through an action channel. Specifically, Alice encodes her message into a quantum action, which subsequently affects the environment state. For example, a quantum measurement at the encoder can induce a state collapse of the environment. In addition, Alice has access to side information. Unlike the classical model, she cannot have a copy of the environment state due to the no-cloning theorem. Instead, she shares entanglement with this environment. We establish an achievable communication rate for reliable message transmission via the quantum action-dependent channel, thereby extending the classical action-dependent framework to the quantum domain.

Figures (2)

• Figure 1: Coding over a quantum action-dependent channel. Here Alice acts as the Action encoder, encoding the message $M$ into an action sequence $G^n$, and the main encoder, encoding the message and side information $S_0^n$ into the channel input $A^n$. The action sequence $G^n$ is fed into the action channel $\mathcal{T}^{\otimes n}_{G \to S S_0}$, which produces the environment state $S^n$ and side-information $S_0^n$ for Alice. The quantum communication channel $\mathcal{N}^{\otimes n}_{SA \to B}$ takes the environment state $S^n$ and input $A^n$, producing the output $B^n$, which is measured by Bob to decode the message.
• Figure 2: Effect of pinching on a quantum state. Left: Matrix representation of $B$. Off-diagonal blocks (regions labeled $C$ and $C^\dagger$) indicate non-commutativity with $A$. Right: After applying the pinching map $\mathcal{E}_A$, the modified state $\mathcal{E}_A(B)$ is block-diagonal in the eigenbasis of $A$ (off-diagonal blocks are zero). Now $\mathcal{E}_A(B)$ commutes with $A$, enabling a measurement comparison of their spectra.

Theorems & Definitions (9)

• Definition 1: Action-Dependent Code
• Definition 2: Achievable Rate
• Theorem 1: Achievable Rate
• Remark 1
• Proposition 2: One-shot error probability
• Lemma 3: see Anshu2020
• proof
• Lemma 4
• proof